Problem: Multiply the following complex numbers: $({-4+3i}) \cdot ({4-5i})$
Answer: Complex numbers are multiplied like any two binomials. First use the distributive property: $ ({-4+3i}) \cdot ({4-5i}) = $ $ ({-4} \cdot {4}) + ({-4} \cdot {-5}i) + ({3}i \cdot {4}) + ({3}i \cdot {-5}i) $ Then simplify the terms: $ (-16) + (20i) + (12i) + (-15 \cdot i^2) $ Imaginary unit multiples can be grouped together. $ -16 + (20 + 12)i - 15i^2 $ After we plug in $i^2 = -1$ , the result becomes $ -16 + (20 + 12)i - (-15) $ The result is simplified: $ (-16 + 15) + (32i) = -1+32i $